Quantum Computation by Adiabatic Evolution

TL;DR

This paper presents a quantum algorithm for solving satisfiability problems using adiabatic evolution, with polynomial-time performance demonstrated in symmetric special cases.

Contribution

It introduces an adiabatic quantum algorithm for satisfiability and analyzes its efficiency in symmetric problem instances.

Findings

Algorithm runs in polynomial time for certain symmetric cases

Performance depends on the energy gap between ground states

General case analysis remains unresolved

Abstract

We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.